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Abstract In this paper I present a historical inquiry on the relationship between physics and mathematics in electromagnetic theory around the nineteenth century. The investigation is within the domain of the history of physics. By essentially following Maxwell’s fundamental aspects of physics mathematics in his A Treatise on Electricity and Magnetism, some epistemological reflections will be put forth, as well as observations regarding the different scientific approaches between Faraday’s Experimental Researches in Electricity and Maxwell’s science.

1 Physics Mathematics from a Physical Standpoint 1.1 Maxwell’s Debate with Faraday Generally, complete biographical and scientific sketches of Faraday and Maxwell are well documented.1 Thus, for the sake of brevity, here I avoid discourse on their biographical accounts. I will rather comment on some chapters of A Treatise on 1

Mainly: Everitt (Everitt), Pearce (Pearce), Williams (Williams), Agassi (1971, 2008), Arianrhod (Arianrhod), Mahon (Mahon), Russel (Russel), Harman (1990, 1998, 2004), Hamilton (2002, 2004), Gooding (Gooding), Gladstone (Gladstone), Meurig (Meurig), Bence (Bence), Tyndall (Tyndall), Baggott (Baggott), Cantor (Cantor), Glazebrook (Glazebrook), Heaviside (Heaviside), Hirshfeld (Hirshfeld), Thompson (Thompson), Tolstoy (Tolstoy), Heilbron (Heilbron), Darrigol (2000). Particularly James’ studies (James) on Faraday’s correspondence are indispensable. R. Pisano () ´ Centre Franc¸ois Vi`ete Epist´ emologie, Historie des Sciences et des Technique, University of Nantes, France [at that time. Currently: Centre Sciences, Soci´et´es, Cultures dans leurs e´ volutions (Scit´e), University of Lille 1, France] Research Centre for the Theory and History of Science, Department of Philosophy, University of West Bohemia in Pilsen, Czech Republic e-mail: [email protected] E. Barbin and R. Pisano (eds.), The Dialectic Relation Between Physics and Mathematics in the XIXth Century, History of Mechanism and Machine Science 16, DOI 10.1007/978-94-007-5380-8 2, © Springer ScienceCBusiness Media Dordrecht 2013

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Electricity and Magnetism that are relevant to the aim of this paper, as well as on Faraday’s different scientific approach in Experimental Researches in Electricity. I will also include some historical–epistemological reflections within physics mathematics. I also found that several of the most fertile methods of research discovered by mathematicians could be much better in terms of ideas derived from Faraday than in their original form.2

1.2 On Modelling and Processes of Reasoning In two volumes3 of A Treatise on Electricity and Magnetism (Maxwell 1873) Maxwell’s main cultural focus was to propose a new view of electromagnetism and the natural world. The book was also intended to be used as a Cambridge text for students taking the Tripos examinations. Maxwell’s new thought is most evident in the final section of A Treatise on Electricity and Magnetism (Maxwell 1873, II, Pt IV). The physics mathematics aspect starts with his use of mathem