Dr Daniel Mitchell of RWTH Aachen specialises in the history and philosophy of mathematics in the nineteenth century. In this talk he considers the gradual liberalization of the kinds of mathematical object and forms of mathematical reasoning viewed as permissible in physical argument. The spread of an “algebraic” mode of mathematical intelligibility into elementary arithmetical pedagogy, experimental physics, and fields of physical practice like telegraphic engineering was slow and difficult. Developed by leading mathematicians from the 1830s onwards, the algebraic mode permitted mathematical operations between non-numerical entities, which, according to James Clerk Maxwell, were indispensable in articulating the relationship between derived and fundamental units in a so-called “absolute” system of measurement.
A watershed event was a clash that took place during 1878 between J D Everett, an acolyte of Maxwell’s, and James Thomson, Lord Kelvin’s brother, over the meaning and algebraic manipulation of the “dimensional” formulae invented by Maxwell for this purpose. Buoyed by dramatic changes to science education in Britain during the 1870s, and the rising economic importance of electrical science and technology during the 1880s, this clash precipitated the emergence of rival “Maxwellian” and “Thomsonian” approaches towards interpreting and applying “dimensional” equations.
What at first looks like a dispute over a seemingly esoteric mathematical tool for unit conversion turns out to concern Everett’s break with arithmetical algebra in the representation and manipulation of physical quantities. This move prompted a vigorous rebuttal from Thomsonian defenders of an orthodox “arithmetical empiricism”, who, for epistemological, semantic and pedagogical reasons, insisted upon retaining physical correlates for mathematical entities and operations. Their resolute stance illustrates a deep gulf in terms of conceptions of mathematical intelligibility in Victorian Britain between leading mathematicians and those who employed mathematics to further experimental or practical goals.