Abstract
The aim of this paper is to argue that there existed relevant interactions between mechanics and geometry during the first half of the nineteenth century, following a path that goes from Gauss to Riemann through Jacobi. By presenting a rich historical context we hope to throw light on the philosophical change of epistemological categories applied by these authors to the fundamental principles of both disciplines (which they came to regard as hypotheses or conventions). We intend to show that presentations of the changing status of the principles of mechanics as a mere epiphenomenon of the emergence of non-Euclidean geometries are inaccurate, that the relations between the two disciplines were richer than what is usually considered in the literature. These claims will be based on historical and philosophical arguments, starting from the fact that disciplinary boundaries at the time were not rigid as we are used today. It is widely known that the main figures we target worked in different areas, which is a first piece of evidence for the plausibility of our main thesis.
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Notes
We are thinking of cases such as those of Gauss, Plücker, or Kummer in Germany, Legendre, Cauchy, or Liouville in France.
It is in these earlier lectures where Jacobi introduced the analytical formulation of mechanics, usually known as the Hamilton–Jacobi formalism.
This has happened in particular when considering the use of conventions by Poincaré: many interpreters have seen the attribution of the notion of convention to the principles of mechanics and physics as a mere extension of his famous geometric conventionalism. Examples of this view are Friedman (1999), Giedymin (1982), or Zahar (2001). However, as some authors have shown recently (Pulte 2000; de Paz 2018), and as we have quoted above, Poincaré is not the first or the only one to use the term ‘convention’ to characterize the principles of mechanics.
On this topic see especially the rich paper Lützen (1995), to which a referee directed our attention. The works of this same author on Liouville and on Hertz are likewise directly relevant.
We should like to mention another study which is also cited in Lützen’s paper. Ziegler (1985) discusses another area of interaction between geometry and mechanics, the geometrical study of the motion of rigid bodies on the basis of methods belonging to projective geometry and line geometry. This was the work of geometers and practitioners of mechanics who were not satisfied with the ‘rarefied’, strongly analytical orientation of Lagrange, Hamilton and Jacobi.
There is also a letter from Jacobi to Gauss where he writes about following considerations from the Disquisitiones Aritmethicae (Jacobi to Gauss, 27.10.1826, in: Jacobi 1881–1891, Vol. 7, 391–392).
In a letter to the Prussian governor, Theodor von Schön, he criticized the emphasis of the philologists on classical education and stated: “the Greeks could learn [...] a hundred times more from us than we could from them. I mean in the great kingdom of truth, mathematics, and in the equally great kingdom of observation, nature” (quoted in Olesko 1991, 52). With these claims, he wanted to support the reform of Prussian education that other professors were asking for in Königsberg.
He studied in Paris, being a member of Fourier’s circle, and brought French mathematical physics to Germany. At Berlin he could not become a full professor because he lacked the required Latin skills. The close friendship of Jacobi and Dirichlet is expressed in a letter of Rebecca, Dirichlet’s wife, to her nephew after Jacobi’s death: “Jacobi died in the night from Tuesday to Wednesday […] His relationship to Dirichlet was so very sweet, the way they sat together for hours, I called it mathematical silence, and the way they certainly did not treat each other gently, and Dirichlet often uttered the bitterest truths to him, and Jacobi understood so well and his great spirit knew to bow before Dirichlet’s great character” (quoted in Bergman et al. 2012, 49).
See the interesting quote from Eisenstein’s autobiography in Wussing (1984). The very young Eisenstein indicated in 1843 that the “new school founded by Gauss, Jacobi and Dirichlet” avoids “long and involved calculation [...] by the use of a brilliant expedient: it comprehends a whole area in a single main idea, and in one stroke presents the final result with utmost elegance” (quoted in Wussing 1984, 270, fn. 81).
This can be seen as the year of his great triumph with the publication of the ‘Theorie der Abelschen Functionen’, and which also brought a small improvement in his salary and working conditions.
Riemann worked on the unification of light and electrodynamics, see Jungnickel et al. (1986, 176–177); one can mention early papers on mathematical physics, namely on Kohlrausch’s measurements, and on Nobili rings (see Archibald 1991). Very relevant too is the paper handed in 1858, but left unpublished: ‘Ein Beitrag zur Electrodynamik’; Riemann wrote: “My discovery of a connection between electricity and light I have submitted here to the Royal Society [the Göttingen Königliche Gesellschaft]” (see Ferreirós 2006a).
According to Betti, whom Riemann told this in conversation, the idea of employing dissections for characterizing the “order of connection” of a surface (closely linked to the Euler characteristic) came to Riemann in the course of a conversation with Gauss about a topic in physics (cf. Weil 1979). Also Listing was inspired by the links Gauss saw between circumstances affecting physical experiments and topological notions. We know nothing about the influence Listing himself may have had upon Riemann.
From the Akten Alfred Clebsch, 1866 writing to von Warnstedt; quoted in Ferreirós (2006a, 67).
Papers from 1829 and 1833 cited in Jungnickel et al. (1986, 49).
The whole passage in this work is quite interesting; p. 81 ends by saying: “According to his intimate conviction [innersten Ansicht], expressed frequently, Gauss regarded the three dimensions of space as a specific characteristic of the human soul; and people who could not understand this, he called in his humorous vein, Boeotians” (von Waltershausen 1856, 81). Compare with Lotze’s views, mentioned in a later footnote (and see Torretti 1978, Sec. 4.2).
As he defined it, topology studies spatial relations according to their “modality”, and not their “quantity” (Listing 1847, 3); such “modal relations” have to do with spatial domains, their relative position and arrangement.
On Gauss and his role in the mathematization of the study of knots, see the work of Epple (1999).
By way of comparison, the contemporary textbook of Georg Carl Justus Ulrich, professor at Göttingen (Lehrbuch der reinen Mathematik, 1836), does not even mention the question of parallels—which is remarkable since the previous generation of professors (Kästner, Klügel) had done quite a lot to revive it.
There is a tendency to forget how important Lambert was in the eighteenth century; see his Theorie der Parallellinien, 1786, and the interesting webpage http://www.kuttaka.org/~JHL/Main.html (Collected Works - Sämtliche Werke Online). For his influence on Gauss, see Abardia et al. (2012), and Rodríguez (2006).
An example from Riemann’s letters to his brother is given by Dedekind (1876, 521): Riemann speaks about the “interconnection between electricity, galvanism, light and gravity”, emphasizing the progress he has made on it; but he is worried “that Gauss has been working on the same topic for years, and has told some friends about the matter, e.g. Weber, under the pledge of secrecy”. In this case his fears were unfounded, since Gauss had nothing similar to his ideas on the question.
Perhaps other members of the Philosophy Faculty, such as the prominent philosopher R. H. Lotze, could have been present. As Dedekind tells us (1876, 517), Riemann made great efforts to present his ideas so that they could be understandable [möglichst verständlich] to “all, including also Faculty members who lacked mathematical education”. (But Lotze was certainly not in agreement with Gauss, he was “a Boeotian” and went so far as calling “scientific freaks” [sic in the 1884 English translation] those who “intimidate” the general public talking about spaces of 4 or 5 dimensions; thus “they make sport of logical distinctions” (Lotze 1884, Sec. 172, p. 173).
Gauss expresses his “joy” to see how easily Bessel had come to coincide with his own views in 1829, and remarks again that so few have an “open sense” for the matter.
Even though he cannot prove it, despite all of his ideas and studies of the geometry of surfaces, and in particular of the pseudosphere (a surface of constant negative curvature, regionally like a non-Euclidean plane). Rigorous proof had to wait until the work of Beltrami.
See Riemann (1854/68, 653, 659–661); on p. 661, Riemann wrote: “If we suppose that bodies exist independently of position, the curvature is everywhere constant, and it then results from astronomical measurements that it cannot be different from zero [...].” It might be that Riemann relied on Bessel’s observations, transmitted through Weber or Listing; for it was Bessel who, in 1838, first measured an annual parallax (0.3136″) for the star 61 Cygni, at a distance from the Earth of 657,000 AU. See Kragh (2012).
This is coherent with his remarks in a review published in 1816, emphasizing the role of the “fructifying, living intuition” (see Ferreirós 2006b, 229), and with the fact that Gauss saw the need for a topological branch of the general theory of magnitudes.
That varied along the century, but at the beginning, according to Bos (1980, 329) in many histories of mathematics mechanics was ranged under “mixed mathematics”. During the nineteenth century the distinction between pure and mixed mathematics will be transformed into pure and applied mathematics (see Epple et al. 2013 for more information about this transformation), and mechanics will be usually included under applied mathematics. An exception is Crelle’s Journal, where in 1826 mechanics was listed as a branch of pure mathematics, in the sense that insofar it is analytic mechanics, it is a branch of analysis (cf. Jungnickel et al. 1986, 27).
This idea, common as it might appear today, was not common at the time. When Neumann implemented mechanics as the introduction to his seminar course in theoretical physics, “his decision […] did not derive from his own educational experiences” (Olesko 1991, 125).
In the original “ad 1 im kleinen Bären”. This could be a reference to observations of a star (numbered 1), which is likely to be Polaris.
For the relevance of the Besselian experiment to Neumann’s conception of Mechanics see Olesko (1991, Ch. 4).
Herbart replaced Kant in Königsberg before moving to Göttingen in 1833 (until his death in 1841). For his influence on Riemann (who emphasized his dissent about metaphysics) see Scholz (1982a), Ferreirós (2006a). For his general relevance in nineteenth century philosophy, which is greater than usually realized, see Beiser (2015).
See also the Herbart texts cited in Scholz (1982a, 421–22). On continuity and its role in Riemann’s mathematics, see Ferreirós (2006a, 79) and Laugwitz (1999). It may be mentioned here that Discrete/Continuous is precisely the first of Riemann’s antinomies, where those concepts are linked in particular with the constitution of space and time.
We mean in particular the idea of manifolds of non-constant curvature; see Scholz (1982b).
See de Paz (2018, 231) for Jacobi’s conception of simplicity and its relation to conventions and to the principle of inertia.
More than a reshaping of a previously existing system, the process can be described as the creation of the scientific disciplines that we have known since then (see Stichweh 1984).
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Acknowledgements
We want to thank two anonymous referees and the editors of this journal for insightful comments. We are very grateful to Warren Schmaus for the English revision of the manuscript. We also thank the research project “The Genesis of Geometric Knowledge” (FFI2017-84524-P) for funding support.
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de Paz, M., Ferreirós, J. From Gauss to Riemann Through Jacobi: Interactions Between the Epistemologies of Geometry and Mechanics?. J Gen Philos Sci 51, 147–172 (2020). https://doi.org/10.1007/s10838-020-09501-x
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DOI: https://doi.org/10.1007/s10838-020-09501-x