In my anti-positivist counterfactual history, here is the first axiom.
Geometry axiom: The world is geometrical.
The most familiar geometry is Euclidean geometry, where the world is R3 and distance squared is given by dx2 + dy2 + dz2. There are other geometries.
Newtonian mechanics is a geometrical theory. Space is represented by R3, with Euclidean metric. That is a geometrical object because of the linear structure and the metric. Lines can be defined as the short distance between points. Plane, circles, triangles, and other geometric objects can be defined.
It is also a geometrical object because there is a large class of transformations that preserve the metric, and hence also the lines and circles determined by the metric. Those transformations are the rotations, reflections, and translations. For example, the transformation (x,y,z) → (x+5,y,z) preserves distances, and takes lines to lines and circles to circles.
Mathematically, a geometry can be defined in terms of some structure like a metric, or the transformations preserving that structure. This view has been known as the Klein Erlangen program since 1872.
The laws of classical equations can be written in geometrical equations like F=ma, where F is the force vector, a is the acceleration vector, and m is the mass. All are functions on Euclidean space. What makes F=ma geometrical is not just that it is defined on a geometrical space, or that vectors are used. The formula is geometrical because all quantities are covariant under the relevant transformations.
Classical mechanics does not specify where you put your coordinate origin (0,0,0), or how the axes are oriented. You can make any choice, and then apply one of the symmetries of Euclidean space. Formulas like F=ma will look the same, and so will physical computations. You can even do a change of coordinates that does not preserve the Euclidean structure, and covariance will automatically dictate the expression of the formula in those new coordinates.
One can think of Euclidean space and F=ma abstractly, where the space has no preferred coordinates, and F=ma is defined on that abstract space. Saying that F=ma is covariant with respect to a change of coordinates is exactly the same as saying F=ma is well-defined on a coordinate-free Euclidean space.
The strongly geometrical character of classical mechanics was confirmed by a theorem of Neother that symmetries are essentially the same as conservation laws. Momentum is conserved because spacetime has a spatial translational symmetry, energy is conserved because of time translation symmetry, and angular momentum is conserved because of rotational symmetry.
Next, we look at geometries that make physical sense.