Abstract
For more than a century the notion of a pre-established harmony between the mathematical and physical sciences has played an important role not only in the rhetoric of mathematicians and theoretical physicists, but also as a doctrine guiding much of their research. Strongly mathematized branches of physics, such as the vortex theory of atoms popular in Victorian Britain, were not unknown in the nineteenth century, but it was only in the environment of fin-de-siècle Germany that the idea of a pre-established harmony really took off and became part of the mathematicians’ ideology. Important historical figures were in this respect David Hilbert, Hermann Minkowski and, somewhat later, Albert Einstein. Roughly similar ideas can be found also among British theorists, among whom Arthur Eddington, Arthur Milne, and Paul Dirac are singled out. Although largely limited to the period 1870–1940, the paper also considers Max Tegmark’s recent hypothesis of the universe (or multiverse) being a one-to-one reflection of mathematical structures.
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Notes
In his Metaphysische Anfangsgründe of 1786 Immanuel Kant (2004: 6–7) famously stated that chemistry could never be a genuine science because its subject matter was intractable to the method of mathematization and deduction from higher principles. “I assert,” Kant wrote, “that in any special doctrine of nature there can be only as much proper science as there is mathematics therein. … [The principles of chemistry] are not receptive to the application of mathematics.” Many years later, the mathematically trained Göttingen physicist Max Born (1920: 382), a former assistant to Hilbert, described the relationship between chemistry and physics in an imperialist rhetoric strikingly similar to Minkowski’s with respect to physics and mathematics: “We realize that we [the quantum physicists] have not yet penetrated very far into the vast territory of chemistry, yet we have travelled far enough to see before us in the distance the passes that must be travelled before physics can impose her laws upon her neighbour science.” Born tended to think of chemistry in a reductionist sense, as an immature field that could only be turned into a proper science by means of mathematical physics—much like Minkowski and Hilbert thought of physics as a relatively immature science in need of an axiomatic structure.
As late as 1950, in a popular exposition of his latest version of a generalized theory of gravitation, Einstein affirmed that the mathematically satisfying structure of his theory was no guarantee that it corresponded to nature. “Experience alone can decide on truth,” he wrote (Einstein 1950: 17).
Wigner (1960) famously suggested that the pervasive usefulness of mathematics in physics and other natural sciences is miraculous and beyond rational comprehension. His view has been discussed by several later authors, including French (2000), Lützen (2011) and Steiner (1998), some of whom argue that the usefulness of mathematics is far from as unreasonable as claimed by Wigner. For a recent discussion that emphasizes the use of Wigner’s puzzle in educational contexts, see Gelfert (2014).
What Milne criticized was the British tradition of applied mathematics, in which ”mathematics [is] employed merely as a tool, as a kind of necessary but objectionable way of salvation.” By contrast, mathematical physics in the German style of theoretical physics were those mathematical activities “which help in the evolution of that embodiment in logical framework of all the data of perception which is the only thing ultimately worth having in science.” He excluded abstract mathematics, with which he meant “pure mathematics in the narrow sense, concerned only with logic, number, form, order, etc.” (cited in Rebsdorf and Kragh 2002: 56). Apparently he did not think that this kind of pure mathematics was relevant to physics.
Dirac (1931: 60). See also Kragh (1990: 208, 272). Other aspects of Dirac’s use of mathematics are dealt with in Bueno (2005). It should be noted that when Dirac spoke of “pure mathematics” he did not have in mind what most mathematicians associate with the term, which are typically branches of mathematics that have an axiomatic foundation. He generally was satisfied with intuitively founded mathematical ideas and at some occasions expressed his lack of interest in axiomatic structures, rigorous methods, and the whole formal game of pure mathematicians. “I believe,” he wrote in 1964, “that the correct line of advance for the future lies in the direction of not striving for mathematical rigor but in getting methods that work in practical examples” (Kragh 1990: 278).
Richard Feynman (1992: 171) expressed himself in similar terms: “It is quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really has nothing to do with the original thing [nature].”
The brief version of the story is that use of the δ-function (which is not a proper function) was made by Oliver Heaviside in the late nineteenth century and that it was introduced independently by Dirac in 1926. Despite its lack of mathematical rigour, with Dirac’s work it became a powerful tool in physics. It was incorporated into the formal framework of mathematics in 1945, when the French mathematician Laurent Schwartz created the theory of distributions. For details on the early history, see Lützen (1982) who traces this kind of function back to Fourier’s 1822 theory of heat. Schwartz (2000: 209–254) recounts in detail of how he invented distributions, a work which was not, however, primarily motivated by a desire to justify mathematically the physicists’ use of the δ-function.
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Acknowledgments
I would like to thank my colleague Henrik Kragh Sørensen for his interest in and comments on this paper.
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Kragh, H. Mathematics and Physics: The Idea of a Pre-Established Harmony. Sci & Educ 24, 515–527 (2015). https://doi.org/10.1007/s11191-014-9724-8
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DOI: https://doi.org/10.1007/s11191-014-9724-8