These same a priori intuitions, the a priori form of all experience, also explain how mathematics is both a priori (non-empirical) and yet has non-trivial content. In short, a priori intuition supplies the non-empirical content of mathematics. Mathematics has a distinctive subject matter, but that subject matter is not provided by some external reality, Platonic or otherwise. Rather, it is provided a priori – by the mind itself. Intuitive space provides much of the a priori synthetic content for geometry (which is Euclidean for Kant); and intuitive time provides the a priori synthetic content for quantitative mathematics. This makes mathematical knowledge both synthetic and a priori. It is synthetic because it is not mere analysis of concepts, and has an intuitive subject matter. It is a priori because its subject matter or content, spatio-temporality, is given a priori by the form of experience.

However, Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a recoil from the inertia of the fictitious fluid. Poincaré performed a Lorentz boost (to order * v* /* c* ) to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow perpetual motion , a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore, he argued that also in this case there has to be another compensating mechanism in the ether.

Wilhelm Wien (1900) assumed (following the works of Thomson, Heaviside, and Searle) that the * entire* mass is of electromagnetic origin, which was formulated in the context that all forces of nature are electromagnetic ones (the "Electromagnetic World View"). Wien stated that, if it is assumed that gravitation is an electromagnetic effect too, then there has to be a proportionality between electromagnetic energy, inertial mass and gravitational mass. [29] In the same paper Henri Poincaré (1900b) found another way of combining the concepts of mass and energy. He recognized that electromagnetic energy behaves like a fictitious fluid with mass density of
m
=
E
/
c
2
{\displaystyle \scriptstyle {m=E/c^{2}}}
(or
E
=
m
c
2
{\displaystyle \scriptstyle {E=mc^{2}}}
) and defined a fictitious electromagnetic momentum as well. However, he arrived at a radiation paradox which was fully explained by Einstein in 1905. [30]