Q: Why exactly is the speed of light constant in vacuum? I know that's what happens, but I want to know why. Relativity simply works under the assumption of light's constant speed, but that doesn't prove it. It's sort of like saying the product of two numbers is equal to the sum of the same two numbers just because 2+2=2x2. A proof requires more than a phenomenon. - Bill
This is an interesting philosophical question. In physics, we don't prove theories to be right, but we do prove theorems about the math used to hold together our theories. Which theories (whole structures, not just little fragmentary claims) are right is in the end determined by mere phenomena. Nobody gave us any book of true assertions, we have to cobble them together out of observation and mathematical logic.
The key logic behind Special Relativity was that Maxwell's equations for electromagnetism looked like exact, universal laws of physics, and their solution gives light waves with a universal speed. Now it was logically possible that those laws were only true in one special reference frame, but by 1905 no experiment (including the famous attempt by Michelson and Morley) provided any evidence that they failed to work in any inertial frame. Einstein showed that there was a logical, consistent framework (Special Relativity) in which Maxwell's equations worked in all inertial frames, and Newton's laws also almost worked for any objects moving slowly with respect to a frame. From this new framework, all sorts of other effects could be derived, and they were all confirmed. Among those many effects are the energy-dependent lifetimes of particles, the exact dynamics of fast-moving particles, the patterns of radiation from accelerating particles, the magnetism-like velocity-dependent term accompanying each fundamental force, etc.
Ultimately, the framework ran into trouble with gravity, and had to be replaced by General Relativity, which in turn probably will ultimately have to be replaced (maybe by something like String Theory) some day.
So in one sense you're right- we don't prove things the way mathematicians do, but instead have to rely a lot on what we actually see. In another sense you're wrong- we aren't generalizing from one isolated fact (like your numerical example), but fitting a huge collection of diverse observations precisely to an extended logical system.