Negotiating Mathematical Intelligibility in the Exact Sciences, ca. 1830-1920: Physical, Empirical, and Formal Conceptions

The nineteenth century witnessed a gradual liberalisation of the mathematical objects and forms of mathematical reasoning permissible in physical argumentation. Historians of physics note, but take for granted, the co-development of the necessary mathematics and its unproblematic application to physical problems. Conversely, historians of mathematics often implicitly presuppose a modern distinction between “pure” and “applied” mathematics, marginalising the influence of the physical concepts that entered into much “pure” research before the turn of the century. Clearly an opportunity is being missed for histories of mathematics and physics to illuminate each other during a crucial period when these fields overlapped and their boundaries were shifting. In this spirit, our panel addresses the reciprocal interactions between algebraic conceptions and methods, and physical or empirical considerations. In particular, it examines the constraints, assumptions and attitudes that requirements for physical or empirical intelligibility engendered among mathematicians and mathematical physicists, while subjecting these notions to historical scrutiny. We develop the idea that these requirements were not necessarily the same, and could be articulated in multiple, sometimes contradictory ways. Hollings contrasts changing attitudes towards abstract or symbolic algebra during the nineteenth and twentieth century and considers why it only gained acceptance during the latter. Mitchell proposes the idea of a “Second Quantification” of physics, in part to capture changes in the representation and manipulation of physical quantities, particularly in connection with measurement. And Falconer assesses the impact of Hamiltonian methods in permitting (meta)physical speculation and concept building by the likes of Maxwell and J. J. Thomson.