How continuity became a convention: Poincaré on the applicability of differential calculus in physics

Marij van Strien

Ghent University

In the early twentieth century, Poincaré argued that it is through convention that we can apply differential calculus in physics. I argue that this was motivated by real concerns about the applicability of differential calculus in physics, which arose around 1900 as a consequence of the idea that the mathematical framework used in physical theories may stand in a non-trivial relationship to physical reality itself. In the late eighteenth and early nineteenth century (especially in the French school of rational mechanics), mechanics was thoroughly mathematized, and was founded on first principles which were regarded as axioms or necessary truths. At the basis of mechanics lay a mathematical framework of Euclidean geometry and continuous variables. However, in the course of the nineteenth century, mechanics developed into an empirical science, that was no longer thought to have the same level of certainty as mathematics. At the same time, mathematics was increasingly regarded as being independent of empirical constraints, which was notably expressed in the development of non-Euclidean geometries, but also in logicism in arithmetic (as developed among others by Frege). As a result of the loosened bond between mathematics and physics, it was no longer evident that the continuum of real numbers that mathematicians worked with could be found in nature. Around 1900, a number of physicists such as Mach, Duhem and Poincaré addressed the issue that we can never measure values with mathematical precision, and that experience does not give us the mathematical continuum. It was at least conceivable for them that quantities in nature only took discrete values. This threatened the applicability of differential calculus in physics, which rests on the assumption that physical quantities take continuous values, which can vary with infinitesimal steps. Around 1900, this assumption was no longer evident; it was in fact denied by Boltzmann, who argued that there are no infinitely small quantities in nature, and that therefore, differential equations in physics are always approximations. In response to such concerns, Poincaré argued that we can take physical quantities in nature to form a mathematical continuum by convention. The convention of continuity is according to him based on considerations of simplicity and the principle of sufficient reason. Furthermore, it is by convention that we take functions in physics to be differentiable, a property which is linked with but not identical to continuity of functions. Poincaré argues that we can safely make these conventions because any function can be approximated as closely as one wishes by a function that is continuous and differentiable. He writes that one can still disagree about whether there is continuity in the actual world, but he himself is not very interested in that issue: it is enough to know that there is an empirical equivalence between continuous and discontinuous descriptions of nature, so that physicists can safely choose to work with differential equations. Nevertheless, making the applicability of differential calculus into a convention entails that the foundations of physical theories are not necessarily a direct reflection of how things are in nature.