Describing and understanding the world in the long nineteenth century
Oberwolfach Reports 41/2020, 2020, 52–53, doi 10.4171/OWR/2020/41

During the 19th century, great strides were made in describing quantitatively and understanding qualitatively the behavior of the natural and social worlds. Making sense of increasingly precise quantitative measurements of familiar natural and social phenomena posed great challenges to scientific imagination, but in addition, the century saw the discovery of new forces and fields, in the image of Oersted’s production of a circular magnetic field by flowing electric current through a wire, and Faraday’s law of electromagnetic induction, to mention just two such novelties. In mid-century, Maxwell, building on Faraday’s notion of “lines of force”, proposed a unification of optics and electrodynamics, in a new theory of the “electromagnetic field”. Although it took another twenty years to win over physicists, Maxwell’s theory eventually opened up broad new horizons for physics and technology. Most notably, the propagation of electromagnetic waves in air was demonstrated by Hertz, giving rise not only to wireless telegraphy, but to two revolutions of the 20th century: broadcast radio and radio astronomy.

Quantitative measurements concerned not only the natural but also the human realm, as the state apparatus collected “statistics”, with which it meant to predict and control populations, from gathering data on birth, death, crime, education level, revenue, and the like. The acquisition of such statistics, combined with the use of mathematical modeling, gave rise, by the end of the century, to “economics”. A new and important domain of application for probability arose in the aftermath of Darwin’s theory of natural selection, which inspired his cousin Francis Galton to examine the statistics of human heredity. In a tangential way, statistical studies of the early 19th century further motivated the introduction of statistical reasoning in physics, a movement marked by the mid-century invention of kinetic gas theory by Maxwell and Boltzmann, and by Henri Poincaré’s new methods of celestial mechanics. At the end of the century, Max Planck put probabilistic arguments to use in order to express his law of black-body radiation, setting the scene for the 20th-century revolution of quantum mechanics.

These two streams, electromagnetic field theory and probability, capture broad swaths of new mathematical thinking about the natural and social worlds in the 19th century. The streams are naturally antagonistic (particle vs. field), introducing a fruitful tension.