Carl Runge (1856-1927) was among the foremost applied mathematicians of his time. He pioneered numerical methods and he put them into a framework of building mathematical models. Runge’s work received controversial interpretations regarding the status of numerics and applied mathematics. In the present paper, I want to argue that this controversy is connected to an increasing tension between engineering and mathematics. In the very late 19th century mathematization of engineering gradually advanced while at the same time education and research in “pure” university mathematics followed a different trajectory.
This led to an increasing gap and to the formation of two different camps regarding the role and position of applied mathematics. The one claimed applied mathematics would bridge the mentioned gap and foster an hierarchically ordered architecture with a flow from pure or theoretical to applied mathematics. The other camp favored an applied mathematics that would help to create an autonomously mathematized engineering science. Based on an analysis of Runge’s work, it will be argued that his position combined traits of both camps.
Three points highlight Runge’s position. First, he designed a general methodology of numerical modeling and analysis rooted in pure mathematics. Although aiming for a general viewpoint, he also took into account the problems raised by abstraction and idealization and he insisted that the construction of numerical models had to respect an array of conditions. Namely, and this is the second point, his conception of mathematical modeling respected available instrumentation and concrete design tasks - any model had to be practically executable for a given purpose with a given instrumentation. Thirdly, he saw precise predictions as the main virtue of mathematical models.