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Geometry, mechanics, and experience: a historico-philosophical musing

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Abstract

Euclidean geometry, statics, and classical mechanics, being in some sense the simplest physical theories based on a full-fledged mathematical apparatus, are well suited to a historico-philosophical analysis of the way in which a physical theory differs from a purely mathematical theory. Through a series of examples including Newton’s Principia and later forms of mechanics, we will identify the interpretive substructure that connects the mathematical apparatus of the theory to the world of experience. This substructure includes models of experiments, models of measurement, and modular connections with partial theories. It evolves during the life of a theory as physicists learn how to apply it in various contexts. It should nevertheless be regarded as an integral part of a genuine physical theory since the theory would otherwise degenerate into pure mathematics.

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Notes

  1. Famous expositions of the semantic view include Suppe (1974, 1989), van Fraassen (1980), and Giere (1988). The structuralist variety of this view (Sneed, 1971; Balzer et al., 1987) is more concerned with interpretive substructures (see Torretti, 1980, Chap. 3). For a recent critical overview of received conceptions of scientific theories, see French (2020).

  2. For a philosophical understanding of mechanics, Ronald Giere’s variety of the semantic view is more relevant than others, for it is based on an inspection of the structure of mechanics treatises. His remarks on the role of theoretical models and exemplars anticipate some of what I have to say on interpretive schemes in the same context (Giere, 1988, Chap. 3). Also relevant are van Fraassen’s comments on Atwood’s machine, regarding the necessity of theory-dependent models of measurement (van Fraassen, 2012, pp. 776–777). However, modular structure is completely ignored in these texts.

  3. Vorms (2011).

  4. For an insightful criticism of these intermodular relations, see Wilson (2013). In “the intricate interwebbing we call ’classical mechanics’,” Mark Wilson sees “as effective a grouping of descriptive tools as man has yet assembled” (ibid. p. 104).

  5. On the conceptual importance of asymptotics, see Batterman (2002).

  6. On these points, see Sklar (1993) and Barberousse (2000).

  7. For details and justifications, see Darrigol (2008, 2014, Chap. 9). This definition of theories does not match any of the options considered in French (2020) (syntactic, semantic, representational, fictional). Indeed, these options are too static and too holistic to include the evolutive, modular character of theories. Hopefully, the proposed definition eludes Steven French’s pessimistic conclusion that “there are no such things as theories.”.

  8. For a telling example of such attention, see Cat (2001).

  9. See Heath and Heiberg (1908), Torretti (1978, pp. 1–9), Vitrac (1994), and Greenberg (2007, Chap. 1).

  10. On Euclid’s optics, see Lejeune (1948) and Simon (1988, pp. 63–72). On the seventeenth-century origins of triangulation surveying, see Heilbron (1993, pp. 185–195).

  11. See Heath and Heiberg (1908) and Roche (1998, Chap. 3).

  12. See Helmholtz (1887) and Darrigol (2003).

  13. On Hilbert’s geometry, see Greenberg (2007, Chap. 3) and Gray (2007, Chap. 23). On analytical geometry and its Cartesian origins, see Torretti (1978, pp. 33–39).

  14. Helmholtz (1887).

  15. See Darrigol (2014, Chap. 2).

  16. Newton (1687), Praefacio ad lectorem, English transl. from Cohen and Whitman (1999, pp. 381–382).

  17. Ibid. p. 382.

  18. Newton (1687, pp. 12–13; 1729, vol. 1, pp. 19–20).

  19. Newton (1687, p. 1).

  20. Newton (1729, book 3, p. 224, cor. 4).

  21. Newton (1687, pp. 2–4; 1729, pp. 2–7).

  22. Newton (1687, pp. 5, 7; 1729, pp. 9, 11–12).

  23. Newton (1687, pp. 5–6, 9–11; 1729, pp. 9–10, 15–18).

  24. Newton (1687, p. 12; 1729, p. 19). Furthermore, the uniformity of inertial motions can only be tested in a relative manner.

  25. Newton (1687, p. 20; 1729, pp. 22–24).

  26. Newton (1687, pp. 14–16, 24–25; 1729, pp. 22–24, 38–40).

  27. Newton (1687, pp. 20–23; 1729, pp. 32–36).

  28. Newton (1687, pp. 23–24; 1729, p. 37).

  29. Newton (1687, p. 420; 1729, pp. 236–237 (aphelions); 1728, pp. 8–10 (implausible force)).

  30. Newton (1729, pp. 221–223).

  31. Duhem (1906, p. 308): “La science physique, c’est un système que l’on doit prendre tout entier ; c’est un organisme dont on ne peut faire fonctionner une partie sans que les parties les plus éloignées de celle-là entrent en jeu, les unes plus, les autres moins, toutes à quelque degré.”

  32. This is a good example of Galilean idealization as discussed in McMullin (1985); nature is being “carved at its joints.”.

  33. Clearly, Newton does not regard statics as a defining module of mechanics (he rather sees it as specializing module), but many of his followers did so, for good reasons (in particular, the second law thus acquires a more direct empirical content). See Darrigol (2020).

  34. Newton (1729, p. 40).

  35. Some of these difficulties are discussed in Wilson (2013).

  36. D’Alembert and le Rond (1743). See Hankins (1970), Firode (2001) and Darrigol (2014, pp. 31–36).

  37. Lagrange (1788). See Fraser (1983).

  38. See Blay (1992).

  39. On variational principles, see Lanczos (1966) and Pulte (1989). On Hamilton’s action, see Hankins (1980, pp. 181–198).

  40. Regarding Thomson and Tait, see Smith and Wise (1989, pp. 270–273, 390–395). Regarding Maxwell, see Siegel (1991 pp. 259–263) and Buchwald (1985, pp. 60–61).

  41. See Darrigol (2014, pp. 62–67), and further reference there.

  42. See Darrigol (2014, pp. 58–62), and further reference there.

  43. See Grattan-Guinness (1984).

  44. On British energeticism, see Smith and Wise (1989). On German energeticism, see Robert Deltete’s introductory essay in Helm (2000).

  45. The latter option would be the one taken in French (2020).

  46. On this interconnection, see Darrigol (2014, pp. 73–74).

  47. Delaunay (1860). Action-angle variables are especially useful when dealing with the periodicity properties of a system, as Niels Bohr and Karl Schwarzschild realized in the old quantum theory: see Nakane (2015).

  48. See Truesdell (1968), Essay V: “Whence the law of moment of momentum?”.

  49. “Note soumise à M. Chasles … à l’appui de la candidature de M. Boussinesq [to a Sorbonne chair],” in Saint–Venant to Boussinesq, 22 April 1876, Bibliothèque de l’Institut.

  50. See Darrigol (2013).

  51. Prandtl (1905). See Eckert (2017).

  52. Kármán (1930) and Prandtl (1931).

  53. See Smith (1998) and Darrigol (2014, pp. 83–89).

  54. The analogy of nature is Newton’s third rule of reasoning (Newton, 1729, p. 203). On the various scales of mechanics and the difficulties in connecting them, see Wilson (2013).

  55. See Darrigol (2018, pp. 17, 192, 212).

  56. On Boltzmann’s introduction of stationary ensembles (under a different name), see Darrigol (2014, pp. 506–511). On Hamilton’s analogy between optics and mechanic, see Hankins (1980, Chap. 4).

  57. van Fraassen (2012, pp. 782, 783) writes: “The crafting of a relationship between theory and phenomena is an interplay of theory, modeling, and experiment during which both the identification of parameters and the physical operations suitable for measuring them are determined” as well as “Empirical grounding is this process of simultaneously, harmoniously extending both the theory and the range of relevant evidence.” The details of this process are left to the reader’s imagination.

  58. “In the beginning was the Act.” Thus speaks Goethe’s Faust in a parody of the biblical “In the beginning was the World.” Helmholtz liked to cite this verse to capture the essence of an interactive theory of perception.

References

  • Balzer, W., Moulines, U., & Sneed, J. (1987). An architectonic for science: The structuralist program. North Holland.

    Book  Google Scholar 

  • Barberousse, A. (2000). La physique face à la probabilité. Vrin.

    Google Scholar 

  • Batterman, R. (2002). The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence. Oxford University Press.

    Google Scholar 

  • Blay, M. (1992). La naissance de la mécanique analytique : la science du mouvement au tournant des XVIIe et XVIIIe siècles. Presses Universitaires de France.

    Google Scholar 

  • Buchwald, J. (1985). From Maxwell to microphysics: Aspects of electromagnetic theory in the last quarter of the nineteenth century. The University of Chicago Press.

    Google Scholar 

  • Cat, J. (2001). On Understanding: Maxwell on the methods of illustration and scientific metaphor. Studies in History and Philosophy of Modern Physics, 32, 395–441.

    Article  Google Scholar 

  • Cohen, B., & Whitman, A. M. (1999). [English translation of Newton 1887, with a guide par Bernard Cohen] The Principia: Mathematical principles of natural philosophy. University of California Press.

  • D’Alembert, Jean le Rond (1743) Traité de dynamique, dans lequel les loix de l’équilibre et du mouvement des corps sont réduites au plus petit nombre possible, et démontrées d’une manière nouvelle, et où l’on donne un principe général pour trouver le mouvement de plusieurs corps qui agissent les uns sur les autres, d’une manière quelconque. Paris: David.

  • Darrigol, O. (2003). Number and measure: Hermann von Helmholtz at the crossroads of mathematics, physics, and psychology. Studies in History and Philosophy of Science, 34, 515–573.

    Article  Google Scholar 

  • Darrigol, O. (2008). The modular structure of physical theories. Synthese, 162, 195–223.

    Article  Google Scholar 

  • Darrigol, O. (2013). For a philosophy of hydrodynamics. In R. Batterman (Ed.), The Oxford handbook of philosophy of physics (pp. 12–42). Oxford University Press.

    Google Scholar 

  • Darrigol, O. (2014). Physics and necessity: Rationalist pursuits from the Cartesian past to the quantum present. Oxford University Press.

    Book  Google Scholar 

  • Darrigol, O. (2018). Atoms, mechanics, and probability: Ludwig Boltzmann’s statistico-mechanical writings–an exegesis. Oxford University Press.

    Book  Google Scholar 

  • Darrigol, O. (2020). Deducing Newton’s second law from relativity principles: A forgotten history. Archive for History of Exact Sciences, 74, 1–43.

    Article  Google Scholar 

  • Delaunay, C. (1860). Théorie du mouvement de la lune (2 vols). Mallet-Bachelier.

  • Duhem, P. (1906). La Théorie physique. Son objet et sa structure. Chevalier & Rivière.

    Google Scholar 

  • Eckert, M. (2017). Ludwig Prandtl and the growth of fluid mechanics in Germany. Comptes Rendus Mécanique, 345, 467–476.

    Article  Google Scholar 

  • Firode, A. (2001). La dynamique de d’Alembert. Vrin.

    Google Scholar 

  • Fraser, C. (1983). Lagrange’s early contributions to the principles and methods of mechanics. Archive for History of Exact Sciences, 28, 197–241.

    Article  Google Scholar 

  • French, S. (2020). There are no such things as theories. Oxford University Press.

    Book  Google Scholar 

  • Giere, R. (1988). Explaining science: A cognitive approach. The University of Chicago Press.

    Book  Google Scholar 

  • Grattan-Guinness, I. (1984). Work of the workers: Advances in engineering mechanics and instruction in France, 1800–1830. Annals of Science, 41, 1–33.

    Article  Google Scholar 

  • Gray, J. (2007). Worlds out of nothing: A course in the history of geometry in the 19th century. Springer.

    Google Scholar 

  • Greenberg, M. J. (2007). Euclidean and non-Euclidean geometries: Development and history (4th ed.). Freeman.

    Google Scholar 

  • Hankins, T. (1970). Jean d’Alembert: Science and the enlightenment. Clarendon Press.

    Google Scholar 

  • Hankins, T. (1980). Sir William Rowan Hamilton. Johns Hopkins University Press.

    Google Scholar 

  • Heath, T. L., & Heiberg, J. L. (1908). The thirteen books of Euclid’s elements translated from the text of Heiberg with introduction and commentary by T. L. Heath (3 vols). Cambridge University Press.

  • Heilbron, J. (1993). Weighing imponderables and other quantitative science around 1800. The University of California Press.

    Book  Google Scholar 

  • Helm, G. F. (2000). The historical development of energetics. Transl. and ed. by Robert Deltete. Springer.

  • Helmholtz, H. (1887) Zählen und Messen, erkenntnistheoretisch betrachtet. In Wissenschaftliche Abhandlungen (vol. 3, pp. 356–391). Barth, 1882–1895.

  • Kármán, T. (1930). Mechanische Ähnlichkeit und Turbulenz. Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, Nachrichten (1930), 58–76.

  • Lagrange, J. L. (1788). Méchanique analitique. Desaint.

  • Lanczos, C. (1966). The variational principles of mechanics (3rd ed.). University of Toronto Press.

  • Lejeune, A. (1948). Euclide et Ptolémée. Deux stades de l’optique géométrique grecque. Bibliothèque de l’Université.

    Google Scholar 

  • McMullin, E. (1985). Galilean idealization. Studies in History and Philosophy of Science, 16, 247–273.

    Article  Google Scholar 

  • Nakane, M. (2015). The origins of the action-angle variables and Bohr’s introduction of them in a paper of 1918. In F. Aaserud & H. Kragh (Eds.), One hundred years of the Bohr atom (Scientia Danica, Series M, Mathematica et physica, vol. 1), pp. 290–309.

  • Newton, I. (1687). Philosophiae naturalis principia mathematica. Streater & Smith.

  • Newton, I. (1728). A treatise of the system of the world [anonymous translation of a Latin manuscript written around 1685, “De motu corporum liber secundus”]. London: Fayram.

  • Newton, I. (1729). The mathematical principles of natural philosophy [Andrew Motte’s translation of the second edition (1726) of Newton 1687] (2 vols.). Benjamin Motte.

  • Prandtl, L. (1905). Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In A. Krazer (ed.), Verhandlungen des dritten internationaler Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904 (Leipzig: Teubner), pp. 484–491.

  • Prandtl, L. (1931). On the rôle of turbulence in technical hydrodynamics. Proceedings of the World Engineering Congress, Tokyo 1929 (Tokyo: Kogakkai), vol. 5, pp. 495–507.

  • Pulte, H. (1989). Das Prinzip der kleinsten Wirkung und die Kraftkonzeptionen der rationalen Mechanik: Eine Untersuchung zur Grundproblematik bei Leonhard Euler, Pierre Louis Moreau de Maupertuis und Joseph Louis Lagrange. Steiner.

    Google Scholar 

  • Roche, J. (1998). The mathematics of measurement: A critical history. Athlone Press.

    Google Scholar 

  • Siegel, D. (1991). Innovation in Maxwell’s electromagnetic theory: Molecular vortices, displacement current, and light. Cambridge University Press.

    Google Scholar 

  • Simon, G. (1988). Le regard, l’être et l’apparence dans l’optique de l’antiquité. Le Seuil.

    Google Scholar 

  • Sklar, L. (1993). Physics and chance: Philosophical issues in the foundations of statistical mechanics. Cambridge University Press.

    Book  Google Scholar 

  • Smith, C. (1998). The science of energy: A cultural history of energy physics in Victorian Britain. Chicago University Press.

    Google Scholar 

  • Smith, C., & Wise, N. (1989). Energy and empire: A biographical study of Lord Kelvin. Cambridge University Press.

    Google Scholar 

  • Sneed, J. (1971). The logical structure of mathematical physics. Reidel.

    Book  Google Scholar 

  • Suppe, F. (1974). The structure of scientific theories. The University of Illinois Press.

    Google Scholar 

  • Suppe, F. (1989). The semantic conception of theories and scientific realism. The University of Illinois Press.

    Google Scholar 

  • Torretti, R. (1978). Philosophy of geometry from Riemann to Poincaré. Reidel.

    Book  Google Scholar 

  • Torretti, R. (1980). Creative understanding: Philosophical reflections on physics. The University of Chicago Press.

    Google Scholar 

  • Truesdell, C. (1968). Essays in the history of mechanics. Springer.

    Book  Google Scholar 

  • van Fraassen, B. (1980). The scientific image. Clarendon Press.

    Book  Google Scholar 

  • van Fraassen, B. (2012). Modeling and measurement: The criterion of empirical grounding. Philosophy of Science, 79, 773–784.

    Article  Google Scholar 

  • Vitrac, B. (1994). Introduction to Euclide, Les éléments, vol. 2, Livres V-VI. Presses Universitaires de France.

  • Vorms, M. (2011). Qu’est-ce qu’une théorie scientifique ? Vuibert.

    Google Scholar 

  • Wilson, M. (2013). What is “classical mechanics” anyway? In R. Batterman (Ed.), The Oxford handbook of philosophy of physics (pp. 43–106). Oxford University Press.

    Google Scholar 

Download references

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    Olivier Darrigol

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Darrigol, O. Geometry, mechanics, and experience: a historico-philosophical musing. Euro Jnl Phil Sci 12, 60 (2022). https://doi.org/10.1007/s13194-022-00491-9

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