This question will not be about affine spaces; rather those are mentioned here as one of many examples.
- A vector space has an underlying set and a field of scalars and an operation of linear combination, satisfying certain axioms. Given scalars $c_1,\ldots,c_n$ and vectors $\vec{v}_1,\ldots,\vec{v}_n$, one gets a vector $c_1\vec{v}_1+\cdots+c_n\vec{v}_n$.
- An affine space has an underlying set and and a field of scalars and an operation of affine combination, satisfying certain axioms. Given scalars $c_1,\ldots,c_n$ satisfying $c_1+\cdots+c_n=1$ and points $\mathbf{p}_1,\ldots,\mathbf{p}_n$, one gets a point $c_1\mathbf{p}_1+\cdots+c_n\mathbf{p}_n$.
- If one chooses any point at all --- call it $\mathbf{0}$ --- in an affine space, it becomes a vector space with that as the zero point if if one defines the linear combination $c_1\mathbf{p}_1+\cdots+c_n\mathbf{p}_n$, with $c_1+\cdots+c_n$ not necessarily equal to $1$, to be the affine combination $$ c_1\mathbf{p}_1 + \cdots + c_n\mathbf{p}_n + (1-c_1-\cdots-c_n)\mathbf{0}. $$
- But one can also define an affine space by saying it has an underlying set $A$ of points and a vector space $V$ that acts transitively on $A$.
So someone who reads the second definition of "affine space" thinks an affine space has more structure than a vector space, because when you discard $A$ and keep $V$, you've got a vector space, but when you have both, you have an affine space. But someone who reads the first definition thinks an affine space has less structure than a vector space because you have to choose one point to be the zero point, and that's additional structure.
Now it seems to me that the idea of an affine space as a vector space that has forgotten its origin conveys the idea more clearly and directly than the $(A,V)$ characterization does.
So a way of making an idea precise may illuminate or obscure the initially imprecise idea that was to be made precise.
That means that there actually is an idea to be made precise, that exists independently of the ways of making it precise that are then developed afterwards.
Has anyone every tried to codify good ways of thinking about those independent imprecise ideas, as they have with precise reasoning in mathematics? Or would that put one in danger of making them precise and thereby causing them to cease to be the independent imprecise ideas that were one's topic?