In every definition of an affine space I see, the affine space is defined as a set $A$ with an associated vector space $V$ with a group action of $V$ on $A$.
But I also see that vector spaces are often identified as "affine spaces with an origin".
This makes me think (/hope) that we there should be some equivalent definition of an affine space that doesn't rely on the concept of a vector space. Are there some axioms of an $n$-dimensional affine space (analogous to the ones for a vector space) that make no reference to vector spaces? If so, would we then be able to show that an affine space equipped with an origin satisfies all of the vector space axioms?
Yes, one can define an affine space over a ground field $\Bbb F$ to be a nonempty set $\Bbb A$ endowed with maps $$\mu: \Bbb A \times \Bbb A \times \Bbb A \to \Bbb A$$ and $$\Lambda: \Bbb F \times \Bbb A \times \Bbb A \to \Bbb A$$ that together satisfy a particular list of reasonable axioms. Informally, we should think of these maps as $$(x, y, z) \mapsto x - y + z$$ and $$(r, x, y) \mapsto x - rx + ry, $$ so that the former encodes "sums relative to a temporarily fixed basepoint" and the latter "scalar multiplication relative to a temporarily fixed basepoint".
There are numerous definitions of affine spaces, many of which are recorded at the nLab, including others that make no reference to vector spaces.