stingc

Actually, science does give an unambiguous relation between space and time. However, it is clearly not true that space and time are exactly the same thing. By rejecting this, you have left the realm of physics, and are talking about pure mathematics. There, you can simply write down whatever (non-contradictory) set of axioms you feel like to get the answer you want.

I don't think that's the answer you wanted, though. Let's go back to physics: While it was always possible to specify events by tacking on a time to the position coordinates, relativity taught us that this is essential. In everyday experience, time is absolute. Once an origin is set, everyone agrees on what 5 seconds into the future means. However, it was eventually discovered that this is not correct. Different observers who are naive to relativity will measure different times. This is not normally noticed because the conditions under which the times will differ significantly are fairly extreme. In any case, the fact remains that time is not absolute in this sense.

However, this still does not justify time a 4th dimension. The reason that this is done is that the observers who disagree on the passage of time will also disagree on lengths. In fact, comparing the (classes of) natural coordinate systems for different observers will show that they are related by something that is essentially a rotation of the temporal and spatial axes. In Newtonian physics, different observers could disagree on their spatial coordinates by a rotation (plus translation), but now we see that different observers must compare their coordinates through a more general sort of rotation that unavoidably involves time. It is therefore very natural to associate time as a 4th coordinate (and not simply an absolute parameter).

Now I'm getting to your question. Rotating spatial axes is easy to imagine because they have the same units. In rotating a time axis into a spatial one, there needs to be conversion factor. That conversion factor happens to be the speed of light, c. It is best to think that light moves at the speed c because of the structure of spacetime, rather than thinking that spacetime is the way it is because of light.

If you don't like all of that, there is an equivalent statement that is more compact. In Newtonian physics, the time difference and the distance between two events are both invariants. They are the same to everyone. Define L^2=x^2+y^2+z^2.

It has since been discovered that L can be measured differently by different people. The quantity that has been shown to be well-defined is L^2=x^2+y^2+z^2-c^2t^2. Again, the speed of light shows up as the natural conversion factor between space and time.