That equation only holds in flat space, and technically it should be expressed in terms of differentials so that it applies to accelerating reference frames as well. These matters aside, the equation only shows how the distance and times between two events are related as seen by various different reference frames. For timelike events, you can always find a reference frame in which both events are separated in time but not in space, and for spacelike events you can always find a frame in which the two events are separated in space but simultaneous in time. The Minkowski metric, which is essentially the formula you have written, gives relationships between space and time for various events as seen by different observers. In no way does it say that one cannot exist without the other.
In general, independent of any physics, there is nothing to say that distance wouldn't exist even if we had no time frame in which to measure it. We just wouldn't be able to perceive it, which is different from saying that the physical thing does not exist in and of itself.
Yes, i know. The equations where shown in this respect, as a simple demonstration that distance and time are invariant. However, what do you mean by ''independant of any physics,'' because, surely its commonsense that we wouldn't have any equations describing distance and time, without a knowledge of physics, and including relativity?
More to my understanding, is that if distance is not real, then myself sitting here, and an australian on the other side of the world, may as well be sitting together, and that doesn't make any sense at all.