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The Dialectic Relation Between Physics and Mathematics in the XIXth Century pp 159–179Cite as

The Relationship Between Physics and Mathematics in the XIXth Century: The Disregarded Birth of a Foundational Pluralism

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Part of the book series: History of Mechanism and Machine Science ((HMMS,volume 16))

Abstract

In my previous historical works I suggested that four scientific choices constitute the foundations of Physics. By means of these choices I will interpret the history of the relationship between Mathematics and Theoretical Physics in the nineteenth century. A particular pair of choices shaped the Newtonian relationship between Mathematics and Physics, which was so efficient in producing new theoretical results that it became a paradigm. In the nineteenth century new formulations of mechanics were made according to different basic choices, so that in theoretical Physics three other pairs of choices began to co-exist with the Newtonian relationship. Very few scientists of that time recognised this pluralism; instead, the community of physicists interpreted the new theories as either mere variations of the dominant one, or loose scientific attitudes to be put aside in order to follow the theoretical progress of the dominant paradigm. But just after the mid-century the pluralism of the relationships between Physics and Mathematics came to the fore again, this time the previous alternative choices shaped new physical theories – thermodynamics, electromagnetism – concerning entirely new fields of phenomena. But this novelty was interpreted as simply a conflict – possibly, a contradiction – between the new basic notions and the old ones; in particular, at the end of the nineteenth century there was a great debate about the theoretical role played by the new notion of energy in contrast with the old notion of force. The persistent lack of awareness of the pluralism of relationships is the reason for both the inconclusiveness of this debate and the dramatic crisis occurring in theoretical Physics from the year 1900. This time the crisis was caused above all by two experimental data (both the quantum hν and the light velocity of light c as the highest possible velocity) which are incompatible with the Newtonian relationship between Mathematics and Physics. Correspondingly, two “revolutionary” theories emerged, again according to the alternative choices to those of this paradigm; both theories required a new relationship with Mathematics, including respectively discrete mathematics and groups.

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Notes

  1. 1.

    Grattan-Guinness (1990), p. 61.

  2. 2.

    It is remarkable that Physics curricula – both at the high schools and at the Universities – share the same quadripartite division of theoretical physics (Drago 2004b). This fact shows, on one hand, ingenuity on the part of physics teachers and, on the other hand, the relevance of the foundational framework suggested by the two options.

  3. 3.

    The correspondence Clarke–Leibniz (Alexander 1956) illustrates a polemic about just this RVM.

  4. 4.

    Some historians (e.g., Guicciardini 1999) considered that Newton’s mathematics made use of a “geometrical method”; yet, he solved problems which are manifestly impossible in Euclidean geometry. Rather, Newton’s mathematical method relied upon geometrical intuition, which Cavalieri and Torricelli had already promoted to an AI technique for obtaining a complete theory of calculus, although a less powerful one than Leibniz’ calculus (Drago 2003a).

  5. 5.

    Its foundations generalised also D’Alembert’s attempt to give a new foundation to mechanics (Hankins 1970, pp 174–176).

  6. 6.

    Carnot (1786, p. 102).

  7. 7.

    Let us remark that in the common language of that time the word “algebraic method” often means “by means of calculus”, since Lagrange, in his Theorie des Fonctions (Lagrange 1797), wanted to reduce the latter to something like an algebra.

  8. 8.

    I leave aside Maupertuis’ mechanics since at that time it received dubious appraisals.

  9. 9.

    Recall Du Bois Raymond’s criticism of Cauchy’s inclusion of AI in the definition of a mathematical limit. However, also Cauchy’s mere restriction of AI met with strong resistance; for instance, in England, Newtonian calculus was not dismissed before the end of the century.

  10. 10.

    Poisson’s rivalry with Lagrange is well known (Duhem 1903, 72). In fact, one main difference between Poisson’s mechanics and Lagrange’s concerned the limit operation from a sum to an integral (Ivi, pp. 81ff), precisely the crucial notion marking the difference between constructive mathematics and classical mathematics.

  11. 11.

    It is not by chance that some scientists following different RPMs from the Newtonian suffered persecution. For example, L. Carnot was excluded from the re-born French Académie des Science and then expelled from the country. Galois was rejected twice at the admission examination for the Ecole Polytéchnique and his revolutionary work was ignored by the academicians; he died very young in mysterious circumstances. At the Ecole des Arts et Métiers the classes of Désormes were regularly attended by police spies (Birembaut 1975). Faraday, Comte, Meyer and almost surely Sadi Carnot suffered mental illness. Poncelet’s projective geometry, non-Euclidean geometries by Lobachevsky, Gauss, Bolyai, Riemann and Bellavitis’ vector calculus suffered academic misfortune.

  12. 12.

    In the same years Lobachevsky stressed his opposition to the use of AI (Lobachevsky 1835–1838, Introduction); he wrote a book for re-founding calculus by means of discrete mathematics only (Lobachevsky 1833).

  13. 13.

    According to general opinion, this interpretation successfully reduces thermodynamics to Newtonian mechanics, while Boltzmann’s basic notions – as shown by a table listing them according to the two rival mechanics, Newton’s and L. Carnot’s, (Drago 2004a; Drago and Saiello 1995, Table 3) – belong to L. Carnot’s alternative mechanics. Moreover, his statistical mechanics, in order to interpret the behaviour of innumerable particles which are subject to mechanical principles (AO), applied a mathematics which was reducible, according to Boltzmann’s views, to an essentially discrete mathematics (PI) (Boltzmann 1896). Hence, Boltzmann worked in the Descartesian RPM.

  14. 14.

    Brunschvicg 1922, 624.

  15. 15.

    Agassi 1969, 463.

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  1. University of Pisa, Pisa, Italy

    Antonino Drago

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  1. BP 92208/FAC SCI TECH, Université de Nantes, rue de la Houssinière 2, Nantes Cedex 3, 4432, France

    Evelyne Barbin

  2. , Faculté des Sciences et des Techniques, Université de Nantes, Rue de la Houssinière 2, Nantes, 44322, France

    Raffaele Pisano

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Drago, A. (2013). The Relationship Between Physics and Mathematics in the XIXth Century: The Disregarded Birth of a Foundational Pluralism. In: Barbin, E., Pisano, R. (eds) The Dialectic Relation Between Physics and Mathematics in the XIXth Century. History of Mechanism and Machine Science, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5380-8_8

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