Abstract
The recent discovery of an indeterministic system in classical mechanics, the Norton dome, has shown that answering the question whether classical mechanics is deterministic can be a complicated matter. In this paper I show that indeterministic systems similar to the Norton dome were already known in the nineteenth century: I discuss four nineteenth century authors who wrote about such systems, namely Poisson, Duhamel, Boussinesq and Bertrand. However, I argue that their discussion of such systems was very different from the contemporary discussion about the Norton dome, because physicists in the nineteenth century conceived of determinism in essentially different ways: whereas in the contemporary literature on determinism in classical physics, determinism is usually taken to be a property of the equations of physics, in the nineteenth century determinism was primarily taken to be a presupposition of theories in physics, and as such it was not necessarily affected by the possible existence of systems such as the Norton dome.
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Notes
Boussinesq (1879a) also mentions Cournot as an author who discussed such indeterministic systems, referring to (Cournot 1841). But in fact, Cournot only discusses the problem of uniqueness of solutions to differential equations on a mathematical level, without relating it to problems in physics. Therefore his discussion does not tell us anything about his ideas on the issue of determinism in physics; for that reason I will not discuss it here.
This term is introduced by Korolev (2007).
The similarity between Poisson and Zinkernagel is also pointed out in Fletcher (2012).
Duhamel (1797–1872) was at this time professor of mathematics at the École Polytechnique and member of the Académie des Sciences. See O’Connor and Robertson (2005a).
Boussinesq (1842–1929) was at that time professor of mathematics at the Faculté des Sciences in Lille (Nye 1976). He is mainly known for his work in hydrodynamics, heat and light.
Boussinesq emphasizes that singularies only arise for “les valeurs très-spéciales de A ou par suite de c 2”, where \(c=r^2\frac{d\theta}{d t}\) (Boussinesq 1879a, p. 104).
Fletcher (2012) mentions that Boussinesq used systems similar to the Norton dome as a basis for a theory about free will, and adds in a footnote: “He did not, however, investigate how ubiquitous such systems might be”. However, Boussinesq put much effort in investigating exactly this issue, and had detailed ideas about the probability with which such systems can be expected to appear in reality.
Bertrand (1822–1900) had been a pupil of Duhamel, and was at this time professor in mathematics at the Collège du France and the École Polytechnique, and secretary of the Académie des Sciences. See O’Connor and Robertson (2005b).
That this can make indeterminism disappear is demonstrated in Zinkernagel (2010) who shows that the difference equation for the dome, contrary to the differential equation, does have a unique solution.
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Acknowledgments
I would like to thank Jos Uffink, John Norton, Dennis Dieks, Samuel Fletcher, Eric Schliesser, Maarten Van Dyck, Bernhard Pos, Marcel Boumans, Thomas Müller (Lausanne University), Alexander Reutlinger and Sylvia Wenmackers for useful comments and discussion. I would also like to thank two anonymous referees for their helpful comments. This work was supported by the Research Foundation Flanders (FWO).
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van Strien, M. The Norton Dome and the Nineteenth Century Foundations of Determinism. J Gen Philos Sci 45, 167–185 (2014). https://doi.org/10.1007/s10838-014-9241-0
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DOI: https://doi.org/10.1007/s10838-014-9241-0