Mathematical physics had become an enterprise devoted to unifying and synthesizing science, seeking agreement between theory and experiment. Poincaré solved the Dirichlet problem using physical methods, then applied it to an electromagnetic problem involving the field inside a cavity, to acoustics through vibrating membranes, and to capillarity through the surfaces of soap bubbles.
One of Poincaré's guiding ideas in physics was that probability played a central role. He extended the idea put forward by Laplace in his philosophical essay on probability, viewing scientific truths as probabilistic approximations.
