special relativity - Can I fix a point in Minkowski space to give it a vector space structure?

I looked up the term Minkowski space on Wikipedia. It said

There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogenous space of the Poincaré group with the Lorentz group as the stabilizer.

In their book Metric Affine Geometry, Snapper and Troyer state on page 59:

It cannot be stressed enough that the affine space $X$ is not a vector space. Its points cannot be added and there is no way to multiply by scalars. No point in $X$ is preferred; they all play the same role. In particular, there is no point in $X$ which makes a better origin for a vector space than any other point.

The situation changes radically if we choose a point $c$ in $X$ and keep it fixed. It is now possible to make $X$ into a left vector space over $k$ by using the one-to-one mapping $f$ from $X$ onto $V$ defined by $f(x) = \overrightarrow{c,x}$ for each $x \in X$. All we do is carry the vector space structure of $V$ over to $X$ by means of the mapping $f$.

So here's my question: As I understand it, it makes sense to think of Minkowski space as an affine space since the basic principle of Special Relativity is that no point in $X$ is a preferred reference frame. But does that mean it is then impossible to "fix" a point in $X$ as Snapper and Troyer say can be done? In other words, is there any physical meaning to the idea of fixing a point in the affine space or is that impossible according to SR?

Obviously I am trying to use a mathematician's idea to interpret what can be done physically with Minkowski space.

Answer

Fixing a point is more or less like fixing a coordinate system on your affine space. Then you can identify $X$ with $V$ as stated in the book, where the fixed point $c\in X$ is mapped to the origin of $V$. In other words, fixing a point $c$ in $X$ is like glueing a copy of $V$ onto $X$ in such a way that $O\in V$ overlaps with $c\in X$. As far as the Lorentz group is considered then the coordinates (i.e. the component of the glued copy of $V$ onto $X$) really behave like vectors, but this is no longer the case under more general transformations (consider for instance translations, or the action of the ray inversion from the conformal group).