Last time we saw how to picture Aristotle's world with space-time diagrams. To review: the positions of objects at any particular time can always be specified relative to the origin and coordinate system of the background space. If Hilde is 10m from the center of the universe, then that's a fact, and whether anyone knows it or not, it's true.
We also saw that we can depict changes in position over an interval of time (i.e. motion) by using space-time diagrams. A stationary object is represented by a vertical line (no change in position per unit time), and angled lines can tell us how fast an object is moving.
In Aristotle's universe there is only one correct way to view such a diagram. There we have a notion of absolute rest, and absolute motion. In the Galilean space-time we'll look at below, we trade absolute motion and position for relative motion and position.
What does this mean? Galileo's idea was that there is no observable difference between being motionless, and moving at a constant velocity. If you were in the hold of a ship on perfectly calm seas, you would have no way of knowing whether you were still tied to the pier, or sailing along. A more modern example might be the experience of sitting on an airliner. In calm, straight flight, there is nothing (other than the sound of the engines perhaps) to tell you that you are moving save for the view out the window.
Motion, in Galilean space-time, is relative---I can only specify my velocity with respect to something else. There is no such thing as absolute velocity (or position for that matter). Intuitively this might seem like bad news---it seems like we lose the ability to state as many facts about the world. That is, according to Aristotle, there's a well-defined answer to any question about where anything is in the universe, and how it's moving. We lose that ability in the new framework.
What facts remain? It turns out that while we lose the ability to say anything about absolute velocity, the relative velocities we measure are invariant. What does this mean? Suppose that you are floating in empty space, with no objects or landmarks that you can see. There is no way for you to determine whether you are moving or not, and indeed, Galileo tells us that there is no absolute fact of the matter. Now suppose from the distance you see another astronaut hurtling past, from left to right in your field of view. Are they moving past you, or are you moving past them? Again, there's no absolute fact of the matter in this case, but that doesn't mean we can't say anything about the motions involved.
We can represent the two astronauts (you and your friend) in a space-time diagram. In Figure 1 let's suppose you are the pentagon, and your friend is the triangle. In this diagram you are represented as being at rest, and your friend is moving with a constant velocity (say, 10 meters per second).
However, if we imagine the situation from the point of view of your friend we can see that their situation is largely the same. They experience someone (you) zooming past them. This can be represented by a transformation of the space-time diagram. I've drawn this in figure 2---notice that here it is the triangle which is depicted at rest, with the pentagon traveling past it.
These two pictures are very different. However, there's an important feature which does not change: the relative distance between the two objects at any horizontal 'slice' of time. A consequence of this is that it is easy for anyone to calculate what each person will measure. In this particular case, if you measure the other astronaut passing by at 10 meters per second, they will measure you passing them at 10 m/s (in the other direction). Motion, in this framework, is only relative, but those relative motions will be objective for any observer.
To sum up, the move to Galilean space-time requires giving up the ability to say certain things definitively---there's no longer any fact-of-the-matter about an object's absolute velocity. But we find there remains some shared facts which remain invariant for everyone who is measuring the system (like the difference between any two object's velocities). Next, as we turn to special relativity we'll find a similar pattern---certain features which seem to be objective (or 'absolute') in Galilean space-time will be relativized in Einstein's theory.