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Firstly, I just want to make sure that I've understood the notions of relative and absolute quantities correctly.

Elementary analysis shows that position and velocity are relative quantities. Indeed, position is clearly relative as two inertial frames $S$ and $S'$ displaced by a constant displacement vector $\mathbb{r}_{0}$ will measure the position of an object to be at $\mathbb{r}$ and $\mathbb{r}'$ respectively, the two positions related by $\mathbb{r}=\mathbb{r}'+\mathbb{r}_{0}$. As these two frames are arbitrary and neither can be distinguished from the other as a preferred absolute rest frame, it must be that position is relative. This argument also holds if the two frames $S$ and $S'$ are in relative motion to one another, related by $\mathbb{r}=\mathbb{r}'+\mathbb{v}t$, where $\mathbb{v}$ is the relative velocity between the two frames. Clearly it follows from this (by differentiating with respect to time) that velocity is also relative.

Now, if I understand it correctly, Newton introduced the notion of absolute space, and thus defining the absolute position and velocity of a given object as their position and velocity measured relative to this frame. Thus these relative quantities defined in the previous paragraph are all related to absolute quantities (that in principal will be the same for all observers at rest relative to absolute space, regardless of where they are located in this space). However, as a result of Galileo's principal of relativity ruling out the existence of a frame at absolute rest, i.e. absence of absolute space, it follows that the concepts of absolute position and velocity do not exist and therefore are truly relative quantities, dependent on the frame that they are measured in.

Secondly, if we consider Maxwell's equations, which are not invariant under Galilean transformations, but we require them to hold in all inertial frames, doesn't it immediately follow that the speed of light has the same constant value in all inertial frames from this assumption (given that Maxwell's equations imply a constant speed of light). Why is it given as an axiom of special relativity?

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Why is it given as an axiom of special relativity?

In a nutshell: because of the Michelson-Morley experiment. This ought to have detected a variation in the speed of light caused by our motion through space. But it didn't. So Einstein reasoned that speed = distance / time, and that if the speed didn't change, your the time did. And then if your time changed, your measurement of distance also had to change. Something like that, Maybe somebody else can express it better or give a reference.

Note that IMHO the postulate "works" because of the wave nature of matter. See The Other Meaning of Special Relativity by Robert Close for what I think is a well-argued article. Again in a nutshell: when you're made of waves along with your rods and clocks, you always measure wave speed to be the same. Because you calibrate your rods and clocks using the motion of waves, then you use them to measure the motion of waves! It's a tautology, see http://arxiv.org/abs/0705.4507. Also note that Einstein abandoned the postulate when he developed general relativity. See for example this and this. The speed of light varies in the room you're in. If it didn't, light wouldn't curve and your pencil wouldn't fall down.

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Secondly, if we consider Maxwell's equations, which are not invariant under Galilean transformations, but we require them to hold in all inertial frames, doesn't it immediately follow that the speed of light has the same constant value in all inertial frames from this assumption (given that Maxwell's equations imply a constant speed of light). Why is it given as an axiom of special relativity?

In all physics frameworks, by which I mean classical mechanics, electrodynamics, thermodynamics and quantum mechanics , one finds similar mathematics, usually differential equations. What separates the frameworks is the region of validity where the model holds.

So Maxwell's equations hold for electromagnetic quantities and the Lorenz transformations ( note Lorenz) are developed within the framework.

The genius of Einstein was in thinking outside the box: He took the proven constant velocity of electromagnetic waves and the kinematics that the Lorenz transformations imposed ,and applied them to massive particles. That is his contribution, that no massive particle can have a speed larger than the velocity of light. The Lorenz transformations for massive particles gave what we now call special relativity .

Not everybody can think outside the box, and many who want to be the new Einstein offer outside the box proposals that are rejected as crank proposals. Einstein had at his fingertips the mathematical formulations of generations of mathematicians and physicists. The combination , background knowledge and out of the box thinking, gave the successful theories he has proposed: his theories/models fitted the data.

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