Here is how I think about the world, pre quantum and pre Planck. First there is the real line, R. R is already pretty spooky—something that you can sub divide indefinitely. Being able to keep subdividing is probably logically simpler than having to stop somewhere, however. The Cartesian product R × R × R = R3 which is a lot like space, both as the Greeks idealized it and as used for most engineering. Two people might agree that the world is R3 without agreeing on the direction that one moves when one of those three numbers change. (We skip over the easier disagreement as to where to origin is.) It is not that the world isn’t R3, but that it is R3 in very many different equivalent ways; sort of too much like R3. There is one thing that R3 doesn’t capture very well—the distance between two points. Our two theorists can agree on how far apart two things are even when they can’t agree on what direction is North, or even up. Pythagoras clarified this nicely with his formula: dist = (dx2 + dy2 + dz2)1/2. (Pythagoras didn’t have the above notation available, but I am sure that he thought as much.) Rotations like:
Galileo made clear that relative motion at speed s, a transformation like
|x' = x + st|
|t' = t,|
That whole theory was quite consistent, but alas wrong. To make a long story short it turns out that space and time are more alike than almost anyone had thought. There is a transformation that mixes time and space like the transformation above mixes x and y. Here it is:
Einstein’s special relativity included stuff also about masses, (E = mc2, etc.), and even electromagnetic fields, but this is all there was to say about space and time. It is clever that the new relative motion equation is more like spatial rotation, and ultimately it is observationally correct. Special relativity’s transformation formula is more nearly like Euclid’s formula, than it is like Galileo’s formula.