The standard story about the development of physics as a separate discipline runs more or less as follows. After the Newtonian revolution of the 17th century, natural science divided into two parts: mechanics on the one hand and experimental science on the other. The latter investigated nature in a mostly qualitative way, whereas mechanics – under the influence of mathematicians such as Lagrange and Legendre – became more and more mathematical, giving rise to new branches of mathematics, in particular calculus.

Towards the end of the 18th century, mathematics started to be used more widely in physics, especially by French scientists like Laplace and Poisson, in accordance with Galileo’s old observation that “the book of nature is written in the language of mathematics”. This trend continued, until, by the end of the 19th century, physics had become well established as a separate discipline, in which physicists performed quantitative experiments and explained the results with the help of theories framed in the language of mathematics.

What is implicit in this story is a remarkable development that took place around the middle of the 19th century: the emergence of theoretical physics as a separate branch of physics and the consequent split of physics into an experimental and a theoretical part. For historians, this development raises two intriguing questions. Why did this split occur and why did it take place at this particular time?

Some time ago, the historians of science Christa Jungnickel and Russell McCormmach made the first attempt to provide a historical analysis of the emergence of theoretical physics, especially in the German-speaking world (see Intellectual Mastery of Nature: Theoretical Physics from Ohm to Einstein University of Chicago Press 1986). They mainly focused on institutional factors, such as the development of German universities, teaching needs and financial matters, and found that separate chairs of theoretical physics started to be created around 1870. The actual origins of theoretical physics, however, seemed to lie some decades earlier, when mathematics entered physics in a serious way.

This all sounds very plausible, but it is precisely the latter point that is contested by Elizabeth Garber in this book, The Language of Physics. Her main thesis is as remarkable as it is simple: much 19th-century work, which up to now has been considered theoretical physics, is in fact mathematics. For a long period, physicists were split personalities, as it were. On the one hand they did experimental work, using mathematics – especially calculus – to quantify their results. On the other hand they took mathematics, for instance differential equations, as the starting point for new investigations that in turn gave rise to new, purely mathematical results that were given little or no physical interpretation.

The latter work, which Garber terms “mathematical physics”, was usually published in mathematics journals. In Garber’s view, this trend persisted until well into the 19th century, when British and German physicists finally broke through existing boundaries and created theoretical physics in its own right by adopting the attitude that mathematics was just a tool to be used within physics and for practical purposes only. They took what they needed, often to the dismay of mathematicians, who watched with horror how the physicists ignored proper mathematical rigor to reach their goal.

A striking example is Paul Dirac’s introduction of his well known delta-function, a mathematical monstrosity, but one that was widely and successfully used in physics. Thus, a certain split between mathematics and physics was created, which has persisted till today. (In recent years, however, the gap seems to have been closing a bit: string theory, for example, has given rise to a number of new results in pure mathematics.)

A characteristic example of the split between mathematics and physics is the difference of opinion between Albert Einstein and the mathematician David Hilbert on the general theory of relativity – an example that Garber briefly discusses in the book. In about 1910 Hilbert had started to work in physics, commenting that “physics was too difficult to leave to the physicists”. Among other things, he took up Einstein’s early work on general relativity. Hilbert eventually published a theory that was similar to Einstein’s final version of general relativity and appeared almost at the same time as Einstein’s work. However, whereas Einstein’s theory was firmly based on physical principles, Hilbert’s was much more an exercise in pure mathematics, based on some dubious physical assumptions. Although Einstein admired Hilbert’s mathematical proficiency, he characterized Hilbert’s physics (in a letter to Hermann Weyl) as “infantile”.

To substantiate her thesis, Garber presents a detailed analysis of the development of the mathematization of physics between 1750 and 1914. She starts in 18th-century France and then moves on to the 19th century, and to Britain and the German states. Her approach is to focus on certain topics – such as the wave equation, elasticity and electrostatics – that characterize the general trend, with the contributions of scientists, such as Laplace, Poisson, Ohm, Stokes, Maxwell, Clausius and Helmholtz, discussed in depth. Institutional developments and social factors are also taken into account where needed.

In my view, Garber argues her case convincingly and I consider this book to be a very valuable addition to the existing literature on the history of modern physics. The book is also well written, although it is not bedtime reading: Garber brings in equations and sometimes intricate technical details if these are needed to clarify a point. However, those who make the effort to follow the argument will gain many new insights and obtain a fresh outlook on the mathematization of physics in the 18th and 19th centuries.