Is there an argument, apart from experiments, that we know this is true?
And if we only know it by experiment, how do we know the experiments are precise enough to conclude this?
Stated differently, what argument is there that prevents us from replacing c in the Lorenz transformations by $c' = c + \epsilon$
We know from Maxwell's equations that the speed of light is constant: $$c=\frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}$$ We know from Galilean Relativity that the laws of motion are the same in all inertial frames. So for each observer to measure the same value of the speed of light, it must be the same no matter how fast you're moving relative to it. If you work out the math, you will get some equations that predict that particles with mass are constrained to move slower than the speed of light. Photons are allowed to travel at the speed of light because they have no mass - meaning that they cannot ever be at rest. Having mass means that you can always find a frame relative to which you are at rest.
