1) The condition of two things being so different in nature as to be unable to
exist together.
2) Logic: (Of two or more propositions) unable to be true simultaneously.

The incompatibility between → Galilean relativity
and Mawxell's theory of → electromagnetism.
Maxwell demonstrated that
electrical and magnetic fields propagate as waves in space.
The propagation speed of these waves in a vacuum is
given by the expression c = (ε_{0}.μ_{0})^{-0.5},
where ε_{0} is the electric → permittivity
and μ_{0} is the magnetic → permeability, both
→ physical constants.
Maxwell noticed that this value corresponds exactly to the
→ speed of light
in vacuum. This implies, however, that the speed of light must also be a universal
constant, just as are the electrical and the magnetic field constants! The
problem is that → Maxwell's equations do not relate
this velocity to an absolute background
and specify no → reference frame
against which it is measured. If we accept that the principle of relativity
not only applies to mechanics, then it must also be true that
Maxwell's equations apply in any → inertial frame,
with the same values for the universal constants. Therefore, the speed of light
should be independent of the movement of its source. This, however, contradicts the
vector addition of velocities, which is a verified principle within
→ Newtonian mechanics.
Einstein was bold enough to conclude that the principle of Newtonian
relativity and Maxwell's theory of electromagnetism are incompatible!
In other words, the → Galilean transformation
and the → Newtonian relativity
principle based on this transformation were wrong. There exists, therefore,
a new relativity principle, → Einsteinian relativity,
for both mechanics and electrodynamics that is based on the
→ Lorentz transformation.