I think it works from the other direction. You don't get to say Pi=3.14 and then suddenly circles; the value of Pi is defined by whatever works for circles. Likewise, the speed of light is derived from whatever makes space and time one thing, thus you can't just assign a value to it. The value comes from the underlying structure of theory of relativity.
That light speed is a constant is a premise, an assumption. It doesn't fall out of any math or other physics relationship, right? We keep measuring C to more decimal places, on the assumption that what we measure here will be the same somewhere else. Which is, necessarily, an untested hypothesis.
> That light speed is a constant is a premise, an assumption. It doesn't fall out of any math or other physics relationship, right?
But it does, it falls out of the geometric relationship between the three space dimensions and the time dimension. In a way, the speed of light can be thought of as the speed of time.
Think of the relationship between space and time as orthogonal, as a right angle -- which makes sense, since the time dimension is at right angles to the three space dimensions. If you move quickly through space, you can't also move quickly through time, and the relationship between your time and space velocities is just what you would expect for an orthogonal relationship:
t' = t √(1-v^2/c^2)
t = time at rest
t' = time at velocity v
c = speed of light
Einstein wrote the above in 1905, then his math teacher (Minkowski) read Einstein's paper, saw the above equation, and realized it meant time was a fourth dimension. Minkowski then famously said, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."
In response, Einstein said, "Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore."
But from that point to 1915, Einstein acquired much more math knowledge and used it to write the General theory.
