of caloric. After the development of classical thermodynamics such formulas were interpreted as specialized forms of energy conservation. This could be extended mathematically, e.g., through Fourier’s diffusion equation, which put conservation in a differential form. The net results were formulas accepted as true independent of the ontological foundations intended to support them. Heat conservation did not depend on accepting caloric atoms as real. Energy conservation was deliberately developed independent of assumptions about the atomic constitution of matter.
As indicated in Part One, the development of both mechanics and electrodynamics in the British tradition were entangled with considerations of ontological foundations: Newton’s hard massy corpuscles, Faraday’s ether, Maxwell’s displacement. Eventually both classical mechanics and classical electrodynamics could be formulated and function in a way that was not dependent on such ontological foundations. This does not imply that the foundations are false, but simply that they are not foundational within the framework of classical physics.
Even as ontological foundations were deconstructed, the co-evlution of mathematical anf physical concepts continued. When Maxwell sought a mathematical form that would fit separate components of electrical and magnetic fields he adapted Hamilton's quaternions, which Thomson had dismissed as useless. After Maxwell's death, two of his successors, Heaviside and Gibbs, independently adapted his work to develop vector analysis. One of the great mathematical achievements of the nineteenth century was the theory of analytic functions of a single complex variable. This was adapted to fluid dynamics and electrodynamics, which feature mutually perpendicular vector fields. The adaption required defining potentials and streamlines in a way that fit the formalism. Hilbert developed a theory of eigenfunctions and sought in vain for an equation that yielded both discrete and continuous eigenvalues. When Schrödinger discovered one, his celebrated wave equation, he relied on the newly published treatise of Courant and Hilbert to develop his wave mechanics. Matrices, group theory, and Lie groups supply further examples of mathematical forms that served to guide the formulation of physical concepts. This coevolution of mathematics and classical physics can be indicated in a table.33
33This is adapted from material in Bochner (1988), and Kline (1972)