Differential and integral calculus, Differential equations, Calculus of variations
Partial differential equations
The confused relation between the relative priority of physical and mathematical concepts may be illustrated by focusing one basic concept, ' continuous', which was of crucial significance both in the development of calculus and in the classical/quantum divide. Newton's development of fluxions, or differential calculus, was rooted in doctrines of the intensification and remission of qualities and the assumption that changes were continuous. Leibnitz accorded functions a more foundational role, but thought of functions in physical terms. In effect the continuity of mathematical forms was justified by the continuity of the qualitative changes, or functional forms, they expressed. The nineteenth century arithmetization of analysis initiated by Cauchy led to a separation of mathematics from such physicalistic foundations and the treatment of mathematical continuity through the familiar - limiting process. If mathematical formulations are to supply a foundation for physical theories, then they cannot rely on physical considerations to justify mathematics. Thus functions, expressed in set-theoretic terms, cannot be assumed to be continuous, or analytic, or to have derivatives, simply because they describe well behaved processes. To guarantee continuity one must cover the set with Borel functions.
Historically, the sloppy mathematics that much of functioning physics utilizes was grounded in obsolete ideas on the foundations of calculus. However, when mathematics is treated as a functional tool, rather than as a foundation for theories, then one can still rely on physicalistic justifications for mathematical formulations. Even after the reform of mathematics, physicists still treated 'continuity' as a physical notion. In his pioneering study of quantities Maxwell (Papers, Vol. II, pp. 215-229) took the continuity of space, time, and motion as an intuitive given. Historically these developments were involved with considerable confusion about the nature of mathematics. Today, they are best viewed as considerations concerning the selection, rather than the justification, of mathematical forms. Thus, the older treatment of matter, fields, and even fluids and gases, assumed that since these are continuous, they can be represented by smooth